By what common fraction does $0.\overline{81}$ exceed $0.81$?
Solution: First, we convert $0.\overline{81}$ to a fraction by the following trick. Let $x=0.\overline{81}$. Then $100x=81.\overline{81}$, so we can subtract:

$$\begin{array}{r r c r@{}l}
&100x &=& 81&.818181\ldots \\
- &x &=& 0&.818181\ldots \\
\hline
&99x &=& 81 &
\end{array}$$

Therefore, $x=\frac{81}{99}=\frac{9}{11}$.

At this point, we could write $0.81$ as $\frac{81}{100}$ and subtract this from $\frac{9}{11}$. However, the following observation will save us some work: \begin{align*}
0.\overline{81} - 0.81 &= 0.818181\ldots - 0.81 \\
&= 0.008181\ldots \\
&= \frac{x}{100}.
\end{align*} Therefore, $$0.\overline{81} - 0.81 = \boxed{\frac{9}{1100}}.$$